The GPS (Global Positioning System) consists, in part, of a constellation of NAVSTAR (Navigation System with Timing And Ranging) spacecraft with transmitters that continuously broadcast L-BAND radio carrier signals on two frequencies. NAVSTAR spacecraft all have near-circular orbits with radial distances that are 4.16 Earth radii from Earth center, with heights of 20182 km above the Earth's surface. With orbit period of 717.957 minutes and inclination to Earth's equator of 55 degrees, each NAVSTAR completes two orbits about the Earth in one sidereal day (a sidereal day is defined by one complete rotation of the Earth relative to the mean locations of a particular ensemble of stars, or to a particular ensemble of quasars. A solar day has 1440 minutes). NAVSTAR radio transmissions are directed towards the Earth.
USER spacecraft employ GPS receivers, and for most effective receipt of NAVSTAR radio transmissions, have orbits with radial distances that are significantly less than 4.16 Earth radii from Earth center. The large existing class of spacecraft in Low Earth Orbit (LEO) have radial distances that are in the neighborhood of 1.1 Earth radii from Earth center, with heights of roughly 600 km above the Earth's surface (a height of 637.81 km goes with radial distance of 1.1000 er). The orbits of USER spacecraft are estimated by processing measurements derived from NAVSTAR radio signals with the aid of GPS receivers. That is, GPS measurements are used for orbit determination of USER spacecraft equipped with GPS receivers.
Also, Earth-fixed GPS receivers are placed at unknown locations and are activated to collect GPS measurements. These are used to estimate the Earth-fixed receiver locations. We refer to this capability as geolocation.
There are two very different types of measurements derived from the NAVSTAR L-BAND radio carrier signals, namely: (i) pseudo-range measurements and (ii) Doppler carrier phase count measurements. Doppler carrier phase count measurements provide Doppler information on the relative velocity of USER spacecraft with respect to NAVSTAR spacecraft. This enables much more precise orbit estimates and geolocation estimates than those that process pseudo-range measurements only. The present invention refers to USER orbit determination and/or geolocation by processing Doppler carrier phase count measurements, not by processing pseudo-range measurements.
Measurement Representations
All methods of precise orbit determination and geolocation map measurement residuals linearly to state estimate corrections. A measurement residual is the difference between a measurement produced by receiver hardware and the estimated representation of the measurement derived from a state estimate—calculated on a computer. The measurement representation is critical to precise orbit determination and geolocation. In one aspect, the present invention refers most significantly to an improvement of the estimated representation of Doppler carrier phase count measurements.
Geodesy and Geophysics
Orbit determination of USER spacecraft, and geolocation of ground-fixed receivers, has been performed for several decades by processing GPS Doppler carrier phase count measurements. This has been particularly relevant for geodesy and geophysics, and very significant progress has been thereby achieved. But the measurement representations currently used in orbit determination and geolocation capabilities for geodesy and geophysics are fraught with serious problems, needlessly so. The Doppler carrier phase measurement is represented as a range measurement with unknown initial range. How can this be?
Every rigorous measurement representation equation for Doppler carrier phase is composed of several additive terms on the right, with the measurement representation on the left. Some terms on the right are characterized by signal, and some are characterized by noise. One of the terms for signal is called range-difference, or delta-range. This difference is: range at the end of the Doppler carrier phase count interval less range at the beginning of the Doppler carrier phase count interval, where range is a measure of distance between NAVSTAR transmitter and USER GPS receiver. One can rearrange the measurement representation equation so that the range-difference appears on the left, and is a function of the Doppler carrier phase measurement and other terms on the right. Algebraically one can then add the unknown range, at the beginning of the Doppler carrier phase count interval, to both sides of the equation. Now the sum of the unknown range on the left, at the beginning of the Doppler carrier phase count interval, and range-difference on the left defines the range at the end of the Doppler carrier phase count interval. That is, the modified equation presents the range at the end of the Doppler carrier phase count interval on the left as a function of the Doppler carrier phase measurement and the unknown range at the beginning of the Doppler carrier phase count interval, together with other terms, on the right. So orbit determination and geolocation for geodesy and geophysics treats the range at the end of the Doppler carrier phase count interval as the measurement, and requires an associated range measurement representation to form a measurement residual. This is how the Doppler carrier phase measurement is represented as a range measurement with unknown initial range. We shall call it the RANGECP measurement representation, with the superscript CP meaning that RANGECP is derived, in part, from a carrier phase measurement.
Further, the carrier phase measurements used for these range representations are overlapping measurements, not sequential measurements. This has a significant detrimental impact for orbit determination and geolocation, and is described below.
Presentation of Estimation Errors
The prior art geodesist presents small constant error magnitudes (i.e., those not derived directly from his state estimate error covariance matrix function) for his estimates of orbits and geolocations using overlapped RANGECP measurements, and has apparently convinced some technical people of their validity. These small error magnitudes usually refer to position components only, a limited subset of any complete state estimate. The argument has been made that if the geodesist is doing so well, then his estimation technique must surely be correct.
An extensive survey of the relevant literature has failed to identify a single analysis of the state estimate error, including a realistic error covariance. In fact, these analyses do not exist because of limitations in the prior art, including but not limited to the use of overlapped carrier phase measurements.
Each and every technique (including prior art methods of batch least squares, sequential filtering, batch filtering, and sequential smoothing) used by the geodesist to calculate precision estimates also calculates an associated state estimate error covariance matrix, or its inverse, that is used intrinsically to calculate the estimates. If these estimates are optimal and appropriately validated estimates, then their associated state estimate error covariance matrices are realistic. Realistic state estimate error covariance matrices would be invaluable for interpretation and use of orbit and geolocation estimates, but they are never presented. They are never presented because they are always significantly unrealistic. This is frequently admitted and is a problem with the prior art. Credible validation techniques in support of realistic state estimate error covariance presentations cannot be found. The present inventors have looked for them and have asked for them in appropriate places, but they apparently do not exist.
GPS Doppler Carrier Phase Count Measurements
NAVSTAR L-BAND radio carrier signals with specified frequencies are generated by atomic clocks on board each NAVSTAR. When received by any GPS receiver, the frequencies of these signals are Doppler shifted according to the deterministic theory of special and general relativity. The Doppler shift is proportional to the velocity of the USER receiver with respect to the velocity of the NAVSTAR transmitter. Each GPS receiver has its own local clock, and it is designed to create a radio signal with approximately the same frequency as that transmitted by each NAVSTAR. The incoming NAVSTAR Doppler shifted signal to each GPS receiver is differenced with the local receiver signal to isolate the Doppler shift and thereby generate a differenced Doppler phase with Doppler frequency. The Doppler phase is counted by the GPS receiver across a time interval [tm; tn] defined by the receiver clock, where tm<tn, and where m and n are time indices.
[00 sec,10 sec], [10 sec; 20 sec], [20 sec; 30 sec], [30 sec; 40 sec], . . . sequential
[00 sec,10 sec], [00 sec; 20 sec], [00 sec; 30 sec], [00 sec; 40 sec], . . . . overlapping
Define τmn=tn−tm. Two illustrative sequences of four time intervals are displayed here, where the first sequence has a constant τmn=10 seconds (the constant sequential value for τmn will be orbit dependant and relatively small as compared to orbit period) and is sequential, but the second sequence is overlapped with non-constant τmn expanding without bound with each measurement. Each overlapped time interval is the concatenation of a sequence of adjacent sequential time intervals.
FIGS. 1A and 1B have general applicability and distinguish the sequential and overlapped Doppler measurement time intervals graphically.
The phase count measurement Nmn across each time interval [tm; tn] is the sum of an integer number of cycles plus a partial cycle. Thus the GPS Doppler carrier phase count measurement Nmn is a rational number (but typically not an integer), and has units of cycles. If the phase count time intervals are sequential, then the Doppler phase count measurements are sequential, non-overlapping, and τmn remains constant and small. But if the time intervals are overlapped, then the Doppler phase count measurements are overlapped, and τmn becomes indefinitely large. Each overlapped measurement is the sum of a sequence of adjacent sequential measurements.
Table 1 (sequential measurements) and Table 2 (overlapping measurements) present related examples (not that many other distinct and useful examples could be constructed and simplifications have been made for purposes of this example; ordinarily all calculations are performed in double precision to 15+decimals and time is represented in the Gregorian Calendar form or Julian Date) of simulated GPS Doppler carrier phase count measurement values for a GPS receiver on a spacecraft in LEO (Low Earth Orbit), referred to the same NAVSTAR transmitter. These tables are coordinated with the time intervals displayed above and with FIGS. 1A-1B. Columns 1, 2, 4, and 7 are the same for each table. Column 1 is time index, Column 2 is time t, Column 4 is range ρ, and Column 7 is the receiver phase count N0nR. Columns 3, 5, and 6 are different for each table due to the distinction in measurement time interval differences τmn. Column 3 is time interval difference τmn, Column 5 is delta-range Δρ, and Column 6 is the measurement: Nmn for sequential measurement and N0n for overlapping measurement.
TABLE 1Sequential Doppler Carrier Phase Measurementsτmnindext (sec)(sec)ρ (m)Δρ (m)Nmn (cy)NOnR (cy)00018632055.190.0010−955.17−5019.4711018631100.02−5019.4710−274.49−1442.4422018630825.53−6461.9110406.082133.9333018631231.61−4327.98101086.395709.0344018632318.001381.05
The time-tag for each measurement is defined here to be the time at end of the Doppler count interval. Example for Table 1: For time interval [tm; tn]=[20 seconds; 30 seconds], and sequential measurement N23=2133.93 cycles, the time-tag t3=30 seconds.
TABLE 2Overlapped Doppler Carrier Phase Measurementsτmnindext (sec)(sec)ρ (m)Δρ (m)NOn (cy)NOnR (cy)00018632055.190.0010−955.17−5019.4711018631100.02−5019.4720−1229.66−6461.9122018630825.53−6461.9130−823.58−4327.9833018631231.61−4327.9840262.811381.0544018632318.001381.05